Approximately Exact Inference for the Common Odds Ratio in Several 2 × 2 Tables

Abstract

The conditional maximum likelihood estimator (MLE) of the common odds ratio in a sequence of independent 2 x 2 tables is known to be superior to the Mantel-Haenszel estimator in terms of asymptotic efficiency and has the further advantage that its exact distribution is known. However, a long-standing barrier to the widespread use of this estimator has been computational intractability; in particular, the calculation of significance levels, confidence sets, and power based on the exact distribution requires fast and efficient algorithms. An important class of such algorithms form the basis of StatXact, a software package able to solve various aspects of the exact inference problem for a sequence of several 2 x 2 tables in real time. We provide an alternative methodology by developing several useful saddlepoint approximations to the exact distribution of the conditional MLE. The approximations are derived from an interesting representation for hypergeometric random variables as a sum of independent Bernoulli random variables and provide fast, accurate calculations of power functions, p values, and confidence sets. The primary computational burden is in determining the roots of a certain polynomial, which need be done numerically only once for each table. Consequently, the required computational effort is typically minimal; for example, all of the examples herein were done using code written by the authors entirely in S-PLUS.

Compstat 2006 - Proceedings in Computational Statistics

“…Hwang and Yang [HY01] presented an optimality theory for mid-P-values in 2×2 contingency tables, showing how this P-value is the expected value of an optimal P-value resulting from a decision-theoretic approach. Strawderman and Wells [SW98] showed that ordinary P-values obtained with higher-order asymptotic methods without continuity corrections for discreteness yield performance similar to that of the mid-P-value.…”

Higher-order approximations for interval estimation in binomial settings

“…One important feature of this work is that all the investigated methods are used without continuity correction; ordinary p-values calculated in this fashion can be viewed as an approximation to the mid-P value; see Pierce and Peters (1999) and Davison and Wang (2002). The difference between comparing an exact approach with a continuity corrected approximation, and comparing the mid-P approach with the approximation is mentioned also in Strawderman and Wells (1998) for the case of several 2 × 2 tables. In general this view entails a trade-off between the control over the Type I error rate and the decrease in the conservativeness of the corresponding confidence intervals.…”

Exact Hypothesis Tests for Log-linear Models withexactLoglinTest

“…Also, highly accurate saddlepoint approximations have been applied with success (Strawderman and Wells 1998).…”